Questions tagged [experimental-mathematics]
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78 questions
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$\pi$ in terms of polygamma
The computer found this, but couldn't prove it.
Let $\psi(n,x)$ denote the polygamma function.
With precision 500 decimal digits we have:
$$ \pi^2 = \frac{1}{4}(15 \psi(1, \frac13) - 3 \psi(1, \...
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7
votes
1
answer
494
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Experimental mathematics: how are floating point equations discovered/converted to exact equations?
the 2005 AMS article/survey on experimental mathematics[1] by Bailey/Borwein mentions many remarkable successes in the field including new formulas for $\pi$ that were discovered via the PSLQ ...
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1
vote
1
answer
250
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Understanding a group of transformations of the plane $\mathbb{Z} \times \mathbb{Z}$
I'm doing some research in visualizing arithmetic sets (resp. properties, resp. sequences of integers). I try to create patterns (in which I hope to observe some symmetries) by injectively mapping $\...
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76
votes
13
answers
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What computational problems would be good proof-of-work problems for cryptocurrency mining?
What computational mathematics problems that could be used as proof-of-work problems for cryptocurrencies? To make this question easier to answer, I want proof-of-work systems that work in ...
- 2,185
18
votes
0
answers
2k
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Distribution of digits of $pq$-adic idempotents (aka "automorphic numbers")
Let $p$ and $q$ be distinct primes. By the ring of $pq$-adic integers I mean the ring $\mathbb{Z}_{pq} := \varprojlim \mathbb{Z}/(pq)^n\mathbb{Z}$ which is obviously isomorphic to $\mathbb{Z}_p \...
CommunityBot
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5
votes
1
answer
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Periods in the trivial extension algebra of the incidence algebra of the divisor lattice
Definition of $C_L$ for people who like number theory:
Let $m$ be a number with prime factorisation $m=p_1^{n_1} ... p_r^{n_r}$ with $n_i>0$. Define $I_m$ to be the incidence algebra of the ...
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115
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3
answers
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The number $\pi$ and summation by $SL(2,\mathbb Z)$
Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality)
Then, we discovered by heuristic arguments and then verified by computer that
$$\...
- 5,063
18
votes
1
answer
727
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Several conjectured identities for polylogarithms
I asked a question at M.SE a couple of years ago about polylogarithms $\!^{[1]}$$\!^{[2]}$$\!^{[3]}$$\!^{[4]}$ where I conjectured
$$720\,\operatorname{Li}_4\!\left(\tfrac12\right)-2160\,\operatorname{...
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2
votes
0
answers
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Conjectures that can be tested with large numbers of Hecke eigenvalues of GSp(4) automorphic forms
As part of my thesis work I have proved Ibukiyama's conjecture implies something about $\mathrm{SO}(5)$ forms associated to certain lattices lifting to $\mathrm{GSp}(4)$ (This was originally a ...
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4
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When Have Numerology and Computational Experimentation Been Successful?
When has numerology been successfully used in math and science? The Monstrous Moonshine conjecture led to a Fields medal for Borcherds. Balmer's formula for hydrogen spectra led to the Bohr model of ...
CommunityBot
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32
votes
1
answer
925
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Strange convergence of Euler's series for $\zeta(2)$
Using Maple to compare $\pi^2$ and the partial sums of $6\sum_{n=0}^{\infty}\frac{1}{n^2}$ I have noticed something that appears strange.
For instance, let $S_{k}=6\sum_{n=0}^{k}\frac{1}{n^2}$ be ...
CommunityBot
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13
votes
1
answer
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Connection between Infinite continued fractions, elliptic integrals and AGM
It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean.
$$C(x) = x + \frac{1^{2}}{2x + \frac{3^{...
CommunityBot
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5
votes
1
answer
1k
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Lines in image; are they significant to prime numbers if so how?
Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where $...
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votes
3
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May $p^3$ divide $(a+b)^p-a^p-b^p$?
Do there exist positive integers $a,b$ and a prime $p>\max(a,b)$ such that $p^3$ divides $(a+b)^p-a^p-b^p$?
The reader of Kvant magazine A. T. Kurgansky asked to prove that such $a,b,p$ do not ...
CommunityBot
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2
votes
0
answers
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On the cardinality of the set of right-truncatable primes
We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime:
\begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ a_{n}a_{n-1}...
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37
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12
answers
3k
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Interesting conjectures "discovered" by computers and proved by humans?
There are notable examples of computers "proving" results discovered by mathematicians, what about the opposite:
Are there interesting conjectures "discovered" by computers and proved by humans?
...
CommunityBot
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10
votes
2
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647
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Periodicity in iterated powers of sin, cos, exp
Given a complex number $z$, consider the sequence
\begin{align*}
a_0 & = 1\\
a_1 & = (cos(1))^z\\
a_n & = (cos(a_{n-1}))^z
\end{align*}
This question is about trying to understand ...
- 2,093
2
votes
0
answers
266
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Experimentation with partial Euler products
Richard Mathar $[1]\& [2]$ shows that
\begin{align}
&\zeta_{2}(s)\equiv\prod_{\Omega(n)=2}^{}\left(\dfrac{1}{1 - n^{s}}\right)^{-1}=\exp \left(\sum _{k=1}^n \frac{P(k s)^2+P(2 k s)}{2 k}\right)...
- 1,903
3
votes
1
answer
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Trying to prove a congruence for Stirling numbers of the second kind
This a repost of a question I asked at Stack Exchange, but I got no answer so far, so I am trying here, even though it may not suit the "research level" requirement.
Proposition: When $n$ and $m$ are ...
CommunityBot
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5
votes
1
answer
484
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Many integral points on quartic models of elliptic curve via differences of squares
Pick fourth power free integer $n$ ($p^4$ doesn't divide $n$).
Represent $n$ as difference of possibly negative integer squares
$n=v_i^2-u_i^2$.
The goal is to find quadratic polynomial with integer ...
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8
votes
2
answers
810
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Modular congruences related to sums of Catalan numbers
I am curious if somebody can be helpful concerning the following
experimental observation:
There exist two rational sequences $\alpha_0,\alpha_1,\dots$ and
$\beta_0,\beta_1,\dots$, both with values ...
CommunityBot
- 1
12
votes
0
answers
605
views
Power series defined by Witt vectors / Teichmüller representatives of p-adics
Let $K$ be $\mathbb{Q}_p$ for some prime $p$ (or more generally an unramified extension $W(\mathbb{F}_q)$ of $\mathbb{Q}_p$). If $\xi \in K$, we can write it in a unique way in the form $\sum a_i p^i$...
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31
votes
6
answers
2k
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Useful tricks in experimental mathematics
There are a few computational tricks which are useful in experimental mathematics.
These tricks are mostly very elementary and often only given as exercices in books.
A typical example is the ...
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25
votes
1
answer
744
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"Harmonacci" recurrence and identities for $\pi$
While playing with something totally irrelevant I stumbled upon the recurrence:
$$a_{n+1} = \frac{1}{a_n} + a_{n-1}$$
It turns out that given $a_0 = 1, a_1 = 1$,
$$lim \frac{a_{2n}}{a_{2n-1}} = \...
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votes
4
answers
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buffon needle experiment [closed]
Hi, what are the "best" values for lenght of needle (l) and distance between paralles (d) for an accurate approximation of pi? Does it have to be l-d-1.0 or ld? Thanx
- 427
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4
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Does the set of happy numbers have a limiting density?
A positive integer $n$ is said to be happy if the sequence
$$n, s(n), s(s(n)), s(s(s(n))), \ldots$$
eventually reaches 1, where $s(n)$ denotes the sum of the squared digits of $n$.
For example, 7 is ...
CommunityBot
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2
votes
1
answer
295
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A determinant involving only cyclotomic factors
Let $\alpha:\mathbb Z\longrightarrow \mathbb Z$ be a quadratic polynomial taking only integral values on the integers and consider the sequence of square-matrices with coefficients
$x^{\alpha(i+j)}$ ...
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7
votes
2
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714
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Infinite product experimental mathematics question.
A while ago I threw the following at a numerical evaluator (in the present case I'm using wolfram alpha)
$\prod_{v=2}^{\infty} \sqrt[v(v-1)]{v} \approx 3.5174872559023696493997936\ldots$
Recently, ...
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